Algebra

USA AIME 1990 1 The increasing sequence 2, 3, 5, 6, 7, 10, 11, . . . consists of all positive integers that are neither the square nor the cube of a positive integer. Find the 500th term of this sequence. 2 Find the value of (52 + 6 ? 43)3/2 ? (52? 6 ? 43)3/2. 3 Let P1 be a regular r-gon and P2 be a regular s-gon (r ? s ? 3) such that each interior angle of P1 is 5958 as large as each interior angle of P2. What?s the largest possible value of s? 4 Find the positive solution to 1 x2 ? 10x? 29 + 1 x2 ? 10x? 45 ? 2 x2 ? 10x? 69 = 0 5 Let n be the smallest positive integer that is a multiple of 75 and has exactly 75 positive integral divisors, including 1 and itself. Find n/75. 6 A biologist wants to calculate the number of fish in a lake. On May 1 she catches a random

USA AIME 1988 1 One commercially available ten-button lock may be opened by depressing ? in any order ? the correct five buttons. The sample shown below has {1, 2, 3, 6, 9} as its combination. Suppose that these locks are redesigned so that sets of as many as nine buttons or as few as one button could serve as combinations. How many additional combinations would this allow? 5 10 4 9 3 8 2 7 1 6 2 For any positive integer k, let f1(k) denote the square of the sum of the digits of k. For n ? 2, let fn(k) = f1(fn?1(k)). Find f1988(11). 3 Find (log2 x)2 if log2(log8 x) = log8(log2 x). 4 Suppose that |xi| < 1 for i = 1, 2, . . . , n. Suppose further that |x1|+ |x2|+ ? ? ?+ |xn| = 19 + |x1 + x2 + ? ? ?+ xn|. What is the smallest possible value of n?

USA AIME 1986 1 What is the sum of the solutions to the equation 4 ? x = 12 7? 4 ? x ? 2 Evaluate the product ( ? 5 + ? 6 + ? 7)(? ? 5 + ? 6 + ? 7)( ? 5? ? 6 + ? 7)( ? 5 + ? 6? ? 7). 3 If tanx + tan y = 25 and cotx + cot y = 30, what is tan(x + y)? 4 Determine 3x4 + 2x5 if x1, x2, x3, x4, and x5 satisfy the system of equations below. 2x1 + x2 + x3 + x4 + x5 = 6 x1 + 2x2 + x3 + x4 + x5 = 12 x1 + x2 + 2x3 + x4 + x5 = 24 x1 + x2 + x3 + 2x4 + x5 = 48 x1 + x2 + x3 + x4 + 2x5 = 96 5 What is that largest positive integer n for which n3 + 100 is divisible by n + 10? 6 The pages of a book are numbered 1 through n. When the page numbers of the book were added, one of the page numbers was mistakenly added twice, resulting in an incorrect sum of

USA AIME 1983 1 Let x, y, and z all exceed 1 and let w be a positive number such that logxw = 24, logy w = 40, and logxyz w = 12. Find logz w. 2 Let f(x) = |x? p|+ |x? 15|+ |x? p? 15|, where 0 < p < 15. Determine the minimum value taken by f(x) for x in the interval p ? x ? 15. 3 What is the product of the real roots of the equation x2 + 18x+ 30 = 2 ? x2 + 18x+ 45? 4 A machine-shop cutting tool has the shape of a notched circle, as shown. The radius of the circle is ? 50 cm, the length of AB is 6 cm, and that of BC is 2 cm. The angle ABC is a right angle. Find the square of the distance (in centimeters) from B to the center of the circle. A B C 5 Suppose that the sum of the squares of two complex numbers x and y is 7 and the sum of